Realising wreath products of cyclic groups as Galois groups. (English) Zbl 0662.12010

Odoni, R. W. K.

Mathematika 35, No. 1, 101-113 (1988).

Let K be a field of characteristic zero, T, X indeterminates algebraically independent over K. For \(n\geq 1\), let k(n)\(\geq 2\) be an integer, \(f_ n(X,T)=X^{k(n)}+T\). Put \(F_ 1(X,T)=f_ 1(X,T)\) and define for \(n\geq 1\), \(F_{n+1}(X,T)=F_ n(f_{n+1}(X,T),T).\)
In theorem 1 the author proves that if \(\overline{K}\) is the algebraic closure of K then the Galois group of \(F_ n(X,T)\) over \(\overline{K}(T)\) is isomorphic to the wreath product \(\Gamma_ n=G_ 1[...[G_ n]...]\) where for each \(i\leq n\), \(G_ i\) is the cyclic group of order k(i) with its natural permutation action on the symbols 1,...,k(i).
As a corollary the author proves that if K is a Hilbertian field containing the k(i)-th roots of 1 for \(i\leq n\) then given \(t>1\) there is a finite Galois extension L over K such that Gal(L/K) is isomorphic to the direct product of t copies of \(\Gamma_ n\).

Reviewer: T.Soundararajan

MSC:

11R32	Galois theory
12F10	Separable extensions, Galois theory
20F29	Representations of groups as automorphism groups of algebraic systems
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures

Keywords:

wreath products of cyclic groups; inverse problem of Galois theory; Galois group; Hilbertian field

References:

[1]	Lang, Algebra (1965)
[2]	Hall, The Theory of Groups pp 81– (1959)
[3]	Grothendieck, Lecture Notes in Maths pp 224– (1971)
[4]	Fried, Field Arithmetic, Ergebnisse der Math., 3\(\deg\) Folge (1986) · doi:10.1007/978-3-662-07216-5
[5]	Lang, Fundamentals of Diophantine Geometry (1983) · doi:10.1007/978-1-4757-1810-2
[6]	DOI: 10.2996/kmj/1138036122 · Zbl 0435.12010 · doi:10.2996/kmj/1138036122
[7]	Šafarevifč, Izv. Akad Nank SSSR, Ser. Mat. 18 pp 26– (1954)
[8]	DOI: 10.1112/plms/s3-51.3.385 · Zbl 0622.12011 · doi:10.1112/plms/s3-51.3.385
[9]	Zariski, Commutative Algebra (1958)